A three-valued function $fV\to \{-1,0,1\}$ defined on the vertices of a graph $G=(V,E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v\in V$, $f\left(N\right(v\left)\right)\ge 1$, where $N\left(v\right)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f\left(V\right)=\sum f\left(v\right)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted ${\gamma }_t^{-}\left(G\right)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph...

In this paper we initiate the study of total restrained domination in graphs. Let $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by ${\gamma }_r^t\left(G\right)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for ${\gamma }_r^t\left(G\right)$ are given in Section 2....

We initiate the study of signed majority total domination in graphs. Let $G=(V,E)$ be a simple graph. For any real valued function $fV\to ℝ$ and $S\subseteq V$, let $f\left(S\right)={\sum }_{v\in S}f\left(v\right)$. A signed majority total dominating function is a function $fV\to \{-1,1\}$ such that $f\left(N\right(v\left)\right)\ge 1$ for at least a half of the vertices $v\in V$. The signed majority total domination number of a graph $G$ is ${\gamma }_{\mathrm{m}aj}^{\mathrm{t}}\left(G\right)=min\{f\left(V\right)\mid f$ is a signed majority total dominating function on $G\}$. We research some properties of the signed majority total domination number of a graph $G$ and obtain a few lower bounds of ${\gamma }_{\mathrm{m}aj}^{\mathrm{t}}\left(G\right)$. ...